Question: Mohamed and Li Jing were asked to find an explicit formula for the sequence $-5\,,-25\,,-125\,,-625,...$, where the first term should be $g(1)$. Mohamed said the formula is $g(n)=-5\cdot5^{{n}}$, and Li Jing said the formula is $g(n)=-5\cdot5^{{n-1}}$. Which one of them is right? Choose 1 answer: Choose 1 answer: (Choice A) A Only Mohamed (Choice B) B Only Li Jing (Choice C) C Both Mohamed and Li Jing (Choice D) D Neither Mohamed nor Li Jing
Answer: In a geometric sequence, the ratio between successive terms is constant. This means that we can move from any term to the next one by multiplying by a constant value. Let's calculate this ratio over the first few terms: $\dfrac{-625}{-125}=\dfrac{-125}{-25}=\dfrac{-25}{-5}={5}$ We see that the constant ratio between successive terms is ${5}$. In other words, we can find any term by starting with the first term and multiplying by ${5}$ repeatedly until we get to the desired term. Let's look at the first few terms expressed as products: $n$ $1$ $2$ $3$ $4$ $g(n)$ ${-5}\cdot\!{5}^{0}$ ${-5}\cdot\!{5}^{1}$ ${-5}\cdot\!{5}^{2}$ ${-5}\cdot\!{5}^{3}$ We can see that every term is the product of the first term, ${-5}$, and a power of the constant ratio, ${5}$. Note that this power is always one less than the term number $n$. This is because the first term is the product of itself and plainly $1$, which is like taking the constant ratio to the zeroth power. Thus, we arrive at the following explicit formula (Note that ${-5}$ is the first term and ${5}$ is the constant ratio): $g(n)={-5}\cdot{5}^{{\,n-1}}$ So Li Jing is definitely right. What about Mohamed? We can see that $g(n)=-5\cdot5^{{\,n}}$ is not a correct formula, because the constant ratio is multiplied one extra time for each term. For instance, according to this formula, the value of the first term would be: $g(1)=-5\cdot5^{{\,1}} = -25$. However, according to our table of values, $g(1)=-5$. So Mohamed is definitely wrong. Only Li Jing got a correct explicit formula.